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System Properties and Free-Response in LTI Systems

This article explores the relationship between LTI views and system properties, as well as the system's free-response and eigenvalues of the state matrix. It also covers the state-space representation and transfer function in the ZVD. Examples and known relationships are discussed, along with PN-Map and frequency response analysis.

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System Properties and Free-Response in LTI Systems

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  1. Systeme-Eingenschaften im Zeit und Frequenzbereich M d Q Tavares , TB425 dqtm@zhaw.ch

  2. x n t f ω S-Plane Z-Plane y SiSy Overview LTI DGl; BSB; ZVD; h(t); g(t); G(ω); G(s) u(t) U(ω) u[n] U(z) y(t) Y(ω) y[n] Y(z) LTD DzGl; BSB; ZVD; g[n]; G(ω);G(z) Control (RT) Telecomm (NTM) SigProc (DSV, ASV) Applied Mathematics (SiSy) Mathematics

  3. Inhalt • Relationship LTI views & System Properties • System Free-Response & Eigenvalues of State Matrix • SS (ZVD) with Laplace Notation (=> Transfer Function) • Overview known relationships • PN-Map and Frequency Response • PN-Map and System Response in time domain

  4. LTI Systeme : Modellierung & Darstellung Physikalisches Prozess Modell-bildung Differential- gleichung Messung (Schrittantwort) Zustands variable Block schaltbild Schrittantwort (analytisch) Impulsantwort Frequenzgang (Bodediagramm) Genauigkeit? Messverfahren Kontinuierliche Systeme Physikalisches Modell Antwort beliebige Eingang (Faltung) Stationäre Antwort zum Schwingungseingang Numerische Simulation nicht genug

  5. State Space (ZVD) System free-response & Eingenvalues of the state matrix State Space (ZVD) representation of LTI system nth-order Free-response, with u(t) =0 to satisfy differential equation, x(t) must have exponential form λi = eigenvalues of state matrix A vi = eigenvectors of state matrix A Solving for λ (non-trivial solution) nth-order polynom characteristic equation

  6. State Space (ZVD) Free-response & Eingenvalues Solving for v (eigenvectors) satisfy this equation with 1 normalised component Given superposition (linearity) principle, if n distinct eigenvalues Apply initial conditions to solve For the αi factors (*) Compare to known free-response of normalised system 2.order with parameters : k, d and ω0 ; or σ and ωe . (*) For larger n, symbolic solution gets complicated, but there are computationally efficient numerical techniques.

  7. State Space (ZVD) Forced response: response to a step input Complete solution (sum) With homogeneous solution And input or stimuli Simplifying for t > 0 The particular solution has form similar to input, and for t>0 derivative equals zero, so: Therefore complete solution Obs.: use initial conditions to solve for αi

  8. State Space (ZVD) System free-response & Eingenvalues of the state matrix State Space (ZVD) representation of LTI system nth-order System response as homogeneous plus particular solution with λi = eigenvalues of state matrix A vi = eigenvectors of state matrix A αi = constants determined by n initial conditions

  9. Inhalt • Relationship LTI views & System Properties • System Free-Response & Eigenvalues of State Matrix • SS (ZVD) with Laplace Notation (=> Transfer Function) • Overview known relationships • PN-Map and Frequency Response • PN-Map and System Response in time domain

  10. State Space (ZVD) ZVD with Laplace Operator & Transfer Function State Space (ZVD) representation of LTI system nth-order & Laplace Transform (with initial conditions equal zero) Manipulating to isolate the transfer function : G(s) = Y(s) / U(s) G(s) Übung 2 : Aufgabe 3

  11. State Space (ZVD) • ZVD with Laplace Operator & Transfer Function (Übertragungsfunktion) • Examples: • Generic 2nd order system • (from BSB exercise with direct form I & II) • - Sallen-Key Butterworth TPF

  12. u(t) y(t) 1/s 1/s -a1 -a0 b1 b2 b0 ZVD & System-Antwort Example 1: Generic 2nd order system direct form II

  13. ZVD & System-Antwort Example 1: Generic 2nd order system Calculating the eigenvalues: and the respective eigenvectors: Gives the complete system response:

  14. ZVD & Übertragungsfunktion Example 1: Generic 2nd order system Übung 2 - Auf 3 Similar but easier with b2 =0

  15. ZVD & Übertragungsfunktion Example 2: Sallen-Key Butterworth TPF For Butterworth pattern:

  16. Inhalt • Relationship LTI views & System Properties • System Free-Response & Eigenvalues of State Matrix • SS (ZVD) with Laplace Notation (=> Transfer Function) • Overview known relationships • PN-Map and Frequency Response • PN-Map and System Response in time domain

  17. LTI Systeme : Modellierung & Darstellung Physikalisches Prozess Modell-bildung Differential- gleichung Messung (Schrittantwort) Zustands variable Block schaltbild Schrittantwort (analytisch) Impulsantwort Frequenzgang (Bodediagramm) Genauigkeit? Messverfahren Kontinuierliche Systeme Physikalisches Modell Antwort beliebige Eingang (Faltung) Stationäre Antwort zum Schwingungseingang Numerische Simulation nicht genug

  18. Relationship among LTI views Skript: Kapitel 6 S. 73

  19. u(t) U(s) y(t) Y(s) LTI Laplace Transformation Inv. Fourier Transformation Convolution with input fct Multiplication with input fct Derivative in time domain Relationship among LTI views • Known relationships • (visualisation : Matlab ltiview) • Differential Equation Frequence Response G(jω) • Differential Equation Transfer Function G(s) • Frequency Response Impulse Response g(t) • Impulse Response System Output y(t) • Transfer Function System Output Y(s) • Step Response h(t) Impulse Response g(t) Fourier Transformation + isolate {Y(jω) / U(jω)} s=jω

  20. Inhalt • Relationship LTI views & System Properties • System Free-Response & Eigenvalues of State Matrix • SS (ZVD) with Laplace Notation (=> Transfer Function) • Overview known relationships • PN-Map and Frequency Response • PN-Map and System Response in time domain

  21. Laplace Transformation : PN-Map Relationship: Transfer Function & Frequency Response (Übertragungsfunktion & Frequenzgang) S-Plane (S-Ebene) Im{s} X Re{s} X Sweep ω from 0 to +∞ , and check for minimum & maximum points

  22. Im{s} Im{s} Im{s} Im{s} Re{s} Re{s} Re{s} Re{s} Laplace Transformation : PN-Map & Freq.Response Examples: determine the filter type (LP, HP, BP, BS) given the PN-Map x x o x x o o x o x Sweep ω from 0 to +∞ , and check for minimum & maximum points

  23. Im{s} Im{s} Im{s} Im{s} Re{s} Re{s} Re{s} Re{s} Laplace Transformation : PN-Map & Freq.Response Examples: determine the filter type (LP, HP, BP, BS) given the PN-Map Tiefpass Hochpass x x o x x o Bandpass Bandsperre o x o x Sweep ω from 0 to +∞ , and check for minimum & maximum points

  24. Im{s} Im{s} Im{s} Im{s} Re{s} Re{s} Re{s} Re{s} Laplace Transformation : PN-Map & Freq.Response Examples: determine the filter type (LP, HP, BP, BS) given the PN-Map Tiefpass Hochpass x x o x x o Bandpass Bandsperre o x o x Sweep ω from 0 to +∞ , and check for minimum & maximum points

  25. Inhalt • Relationship LTI views & System Properties • System Free-Response & Eigenvalues of State Matrix • SS (ZVD) with Laplace Notation (=> Transfer Function) • Overview known relationships • PN-Map and Frequency Response • PN-Map and System Response in time domain

  26. Pol-Nullstellen-Darstellung Beispiel 1

  27. Pol-Nullstellen-Darstellung Beispiel 2

  28. Pol-Nullstellen-Darstellung Beispiel 2

  29. Pol-Nullstellen-Darstellung Beispiel 3

  30. Pol-Nullstellen-Darstellung Beispiel 3

  31. Pol-Nullstellen-Darstellung Erkenntnisse

  32. Pol-Nullstellen-Darstellung Erkenntnisse

  33. Pol-Nullstellen-Darstellung Erkenntnisse

  34. Pol-Nullstellen-Darstellung Erkenntnisse

  35. Pol-Nullstellen-Darstellung Erkenntnisse

  36. Pol-Nullstellen-Darstellung Beispiel 4

  37. Pol-Nullstellen-Darstellung Beispiel 4

  38. Pol-Nullstellen-Darstellung Beispiel 4

  39. Pol-Nullstellen-Darstellung Beispiel 4

  40. Pol-Nullstellen-Darstellung Beispiel 4

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